RENDICONTI
DEL
S
EMINARIO
M
ATEMATICO
Universit`a e Politecnico di Torino
Control Theory and its Applications
CONTENTS
A. Agrachev, Compactness for Sub-Riemannian Length-minimizers and Subanalyticity . . 1 M. Bardi, S. Bottacin, On the Dirichlet problem for nonlinear degenerate elliptic
equa-tions and applicaequa-tions to optimal control . . . 13 R. M. Bianchini, High Order Necessary Optimality Conditions . . . 41 U. Boscain, B. Piccoli, Geometric Control Approach To Synthesis Theory . . . 53 P. Brandi, A. Salvadori, On measure differential inclusions in optimal control theory . . . . 69 A. Bressan, Singularities of Stabilizing Feedbacks . . . 87 F. Bucci, The non-standard LQR problem for boundary control systems . . . 105 F. Ceragioli, External Stabilization of Discontinuos Systems and Nonsmooth Control
Lya-punov-like Functions . . . 115 L. Pandolfi, On the solutions of the Dissipation Inequality . . . 123 F. Rampazzo, C. Sartori, On perturbations of minimum problems with unbounded controls 133
Rend. Sem. Mat. Univ. Pol. Torino Vol. 56, 4 (1998)
A. Agrachev
COMPACTNESS FOR SUB-RIEMANNIAN
LENGTH-MINIMIZERS AND SUBANALYTICITY
Abstract.
We establish compactness properties for sets of length-minimizing admissi-ble paths of a prescribed small length. This implies subanayticity of small sub-Riemannian balls for a wide class of real-analytic sub-sub-Riemannian structures: for any structure without abnormal minimizers and for many structures without strictly abnormal minimizers.
1. Introduction
Let M be a C∞Riemannian manifold, dim M = n. A distribution on M is a smooth linear subbundle1of the tangent bundle T M. We denote by1qthe fiber of1at q∈M;1q⊂TqM.
A number k=dim1qis the rank of the distribution. We assume that 1<k<n. The restriction
of the Riemannian structure to1is a sub-Riemannian structure.
Lipschitzian integral curves of the distribution 1are called admissible paths; these are Lipschitzian curves t7→q(t), t∈[0,1], such thatq˙(t)∈1q(t)for almost all t .
We fix a point q0 ∈ M and study only admissible paths started from this point, i.e. we
impose the initial condition q(0)=q0. Sections of the linear bundle1are smooth vector fields;
iterated Lie brackets of these vector fields define a flag
1q0 ⊂1
2
q0 ⊂ · · · ⊂1
m
q0· · · ⊂TqM
in the following way:
1mq0=span{[X1,[X2,[. . . ,Xm]. . .](q0): Xi(q)∈1q, i=1, . . . ,m, q∈M}.
A distribution1is bracket generating at q0if1mq
0 = Tq0M for some m > 0. If1is bracket
generating, then according to a classical Rashevski-Chow theorem (see [15, 22]) there exist ad-missible paths connecting q0with any point of an open neighborhood of q0. Moreover, applying a general existence theorem for optimal controls [16] one obtains that for any q1from a small
enough neighborhood of q0there exists a shortest admissible path connecting q0with q1. The
length of this shortest path is the sub-Riemannian or Carnot-Caratheodory distance between q0
and q1.
For the rest of the paper we assume that1is bracket generating at the given initial point q0. We denote byρ(q)the sub-Riemannian distance between q0and q. It follows from the
Rashevsky-Chow theorem thatρ is a continuous function defined on a neighborhood of q0. Moreover,ρis H¨older-continuous with the H¨older exponent m1, where1mq0 = Tq0M. A
sub-Riemannian sphere S(r)is the set of all points at sub-Riemannian distance r from q0, S(r)=
ρ−1(r).
In contrast to the Riemannian distance, the sub-Riemannian distanceρis never smooth in a punctured neighborhood of q0(see Theorem 1) and the main motivation for this research is
to understand regularity properties ofρ. In the Riemannian case, where all paths are available, the set of shortest paths connecting q0with the sphere of a small radius r is parametrized by the
points of the sphere. This is not true for the set of shortests admissible paths connecting q0with the sub-Riemannian sphere S(r). The structure of the last set may be rather complicated; we show that this set is at least compact in H1-topology (Theorem 2). The situation is much simpler if no one among so called abnormal geodesics of length r connect q0with S(r). In the last
case, the mentioned set of shortests admissible paths can be parametrized by a compact part of a cylinder Sk−1× n−k(Theorem 3). In Theorem 4 we recall an efficient necessary condition for
a length r admissible path to be a shortest one. In Theorem 5 we state a result, which is similar to that of Theorem 3 but more efficient and admitting nonstrictly abnormal geodesics as well.
We apply all mentioned results to the case of real-analytic M and1. The main problem here is to know whether the distance functionρis subanalytic. Positive results for some special classes of distributions were obtained in [8, 17, 19, 20, 23] and the first counterexample was described in [10] (see [13, 14] for further examples and for study of the “transcendence” ofρ).
Both positive results and the counterexamples gave an indication that the problem is inti-mately related to the existence of abnormal length-minimizers. Corollaries 2, 3, 4 below make this statement a well-established fact: they show very clear that only abnormal length-minimizers may destroy subanalyticity ofρout of q0.
What remains? The situation with subanalyticity in a whole neighborhood including q0is not yet clarified. This subanalyticity is known only for a rather special type of distributions (the best result is stated in [20]). Another problem is to pass from examples to general statements for sub-Riemannian structures with abnormal length-minimizers. Such length-minimizers are exclusive for rank k ≥ 3 distributions (see discussion at the end of the paper) and typical for rank 2 distributions (see [7, 21, 24]). A natural conjecture is:
If k=2 and12q
0 6=1
3
q0, thenρis not subanalytic.
2. Geodesics
We are working in a small neighborhood Oq0 of q0∈ M, where we fix an orthonormal frame
X1, . . . ,Xk ∈Vect M of the sub-Riemannian structure under consideration. Admissible paths
are thus solutions to the differential equations
˙ q=
k
X
i=1
ui(t)Xi(q), q∈ Oq0, q(0)=q0,
(1)
where u=(u1(·), . . . ,uk(·))∈Lk2[0,1]. Belowkuk =R01Pik=0u2i(t)dt
1/2
is the norm in Lk2[0,1]. We also setkq(·)k = kuk, where q(·)is the solution to (1). Let
Ur = {u∈Lk2[0,1] :kuk =r}
be the sphere of radius r in Lk2[0,1]. Solutions to (1) are defined for all t ∈[0,1], if u belongs to the sphere of a small enough radius r . In this paper we take u only from such spheres without
special mentioning. The length l(q(·))= R01
Pk
i=1u2i(t)
1/2
Compactness for Sub-Riemannian 3
the inequality
l(q(·))≤ kq(·)k =r.
(2)
The length doesn’t depend on the parametrization of the curve while the normkukdepends. We say that u and q(·)are normalized ifPki=1u2i(t)doesn’t depend on t . For normalized u, and only for them, inequality (2) becoms equality.
REMARK1. The notationskq(·)kand l(q(·))reflect the fact that these quantities do not depend on the choice of the orthonormal frame X1, . . . ,Xkand are characteristics of the trajec-tory q(·)rather than the control u. L2-topology in the space of controls is H1-topology in the space of trajectories.
We consider the endpoint mapping f : u7→q(1). This is a well-defined smooth mapping of a neighborhood of the origin of Lk2[0,1] into M. We set fr = f
Ur. Critical points of the mapping fr : Ur →M are called extremal controls and correspondent solutions to the equation
(1) are called extremal trajectories or geodesics.
An extremal control u and the correspondent geodesic q(·)are regular if u is a regular point of f ; otherwise they are singular or abnormal.
Let Crbe the set of normalized critical points of fr; in other words, Cris the set of
normal-ized extremal controls of the length r . It is easy to check that fr−1(S(r))⊂Cr. Indeed, among
all admissible curves of the length no greater than r only geodesics of the length exactly r can reach the sub-Riemannian sphere S(r). Controls u∈ fr−1(S(r))and correspondent geodesics
are called minimal.
Let Duf : Lk2[0,1]→Tf(u)M be the differential of f at u. Extremal controls (and only them) satisfy the equation
λDuf =νu
(3)
with some “Lagrange multipliers”λ∈Tf∗(u)M\0,ν∈ . HereλDuf is the composition of the
linear mapping Duf and the linear formλ: Tf(u)M→ , i.e.(λDuf)∈Lk2[0,1]∗=Lk2[0,1].
We haveν 6=0 for regular extremal controls, while for abnormal controlsνcan be taken 0. In principle, abnormal controls may admit Lagrange multipliers with both zero and nonzeroν. If it is not the case, then the control and the geodesic are called strictly abnormal.
Pontryagin maximum principle gives an efficient way to solve equation (3), i.e. to find ex-tremal controls and Lagrange multipliers. A coordinate free formulation of the maximum princi-ple uses the canonical symprinci-plectic structure on the cotangent bundle T∗M. The symplectic struc-ture associates a Hamiltonian vector fieldaE∈Vect T∗M to any smooth function a : T∗M→ (see [11] for the introduction to symplectic methods).
We define the functions hi, i=1, . . . ,k,and h on T∗M by the formulas hi(ψ )= hψ,Xi(q)i, h(ψ )=
1 2
k
X
i=1
h2i(ψ ) , ∀q∈M, ψ∈Tq∗M.
Pontryagin maximum principle implies the following
ψ (t), 0≤t≤1, to the system of differential and pointwise equations ˙
ψ=
k
X
i=1
ui(t)hEi(ψ ) , hi(ψ (t))=νui(t)
(4)
with boundary conditionsψ (0)∈Tq∗
0M, ψ (1)=λ.
Here(ψ (t), ν)are Lagrange multipliers for the extremal control ut :τ 7→t u(tτ ); in other
words,ψ (t)Dut f =νut.
Note that abnormal geodesics remain to be geodesics after an arbitrary reparametrization, while regular geodesics are automatically normalized. We say that a geodesic is quasi-regular if it is normalized and is not strictly abnormal. Settingν=1 we obtain a simple description of all quasi-regular geodesics.
COROLLARY1. Quasi-regular geodesics are exactly projections to M of the solutions to the differential equationψ˙ = Eh(ψ )with initial conditionsψ (0) ∈ Tq∗0M. If h(ψ (0))is small enough, then such a solution exists (i.e. is defined on the whole segment [0,1]). The length of the geodesic equals√2h(ψ (0))and the Lagrange multiplierλ=ψ (1).
The next result demonstrates a sharp difference between Riemannian and sub-Riemannian distance functions.
THEOREM1. Any neighbourhood of q0in M contains a point q6=q0, where the distance
functionρis not continuously differentiable.
This theorem is a kind of folklore; everybody agrees it is true but I have never seen the proof. What follows is a sketch of the proof.
Supposeρ is continuously differentiable out of q0. Take a minimal geodesic q(·)of the
length r . Thenτ7→q(tτ )is a minimal geodesic of the length tr for any t∈[0,1] and we have
ρ(q(t)) ≡r t ; hencehdq(t)ρ ,q˙(t)i =r . Since any point of a neighborhood of q0belongs to some minimal geodesic, we obtain thatρhas no critical points in the punctured neighborhood. In particular, the spheres S(r)=ρ−1(r)are C1-hypersurfaces in M. Moreover, S(r)=∂f(Ur);
hence dq(1)ρ
Dufr = 0 and we obtain the equality dq(1)ρ
Duf = 1ru, where u is the
extremal control associated with q(·). Hence q(·)is the projection to M of the solution to the equationψ˙ = Eh(ψ )with the boundary conditionψ (1)=r dq(1)ρ. Moreover, we easily conclude thatψ (t)=r dq(t)ρand come to the equation
˙ q(t)=r
k
X
i=1
hdq(t)ρ ,Xi(q(t))iXi(q(t)) .
For the rest of the proof we fix local coordinates in a neighborhood of q0. We are going to prove
that the vector field V(q) = rPki=1hdqρ ,Xi(q)iXi(q), q 6= q0, has index 1 at its isolated
singularity q0. Let Bε = {q ∈ n : |q−q0| ≤ ε}be a so small ball thatρ(q) < r2,∀q ∈
Bε. Let s 7→ q(s;qε)be the solution to the equation q˙ = V(q)with the initial condition
q(0;qε)=qε ∈ Bε. Then q(r2;qε)6∈ Bε. In particular, the vector field Wεon Bεdefined by the formula W(qε)=q(2r;qε)−qεlooks “outward” and has index 1. The family of the fields
Vs(qε)= 1s(q(s;qε)−qε), 0 ≤s ≤ r2provides a homotopy of V
Bε and
r
2W , hence V has
Compactness for Sub-Riemannian 5
On the other hand, the field V is a linear combination of X1, . . . ,Xk and takes its values
near the k-dimensional subspace span{X1(q0), . . . ,Xk(q0)}. Such a field must have index 0 at
q0. This contradiction completes the proof.
Corollary 1 gives us a parametrization of the space of quasi-regular geodesics by the poins of an open subset9of Tq∗0M. Namely,9consists ofψ0∈Tq∗0M such that the solutionψ (t)to
the equationψ˙ = Eh(ψ )with the initial conditionψ (0)= ψ0is defined for all t ∈[0,1]. The composition of this parametrization with the endpoint mapping f is the exponential mapping
:9→M. Thus (ψ (0))=π(ψ (1)), whereπ : T∗M→M is the canonical projection. The space of quasi-regular geodesics of a small enough length r are parametrized by the points of the manifold H(r) = h−1(r22)∩Tq∗
0M ⊂ 9. Clearly, H(r)is diffeomorphic to
n−k×Sk−1and H(sr)=s H(r)for any nonnegative s.
All results about subanalyticity of the distance functionρare based on the following state-ment. As usually, the distances r are assumed to be small enough.
PROPOSITION2. Let M and the sub-Riemannian structure be real-analytic. Suppose that there exists a compact K ⊂h−1(12)∩Tq∗0M such that S(r)⊂ (r K),∀r∈(r0,r1). Thenρis
subanalytic onρ−1((r0,r1)).
Proof. It follows from our assumptions and Corollary 1 that
ρ(q)=min{r :ψ∈K, (rψ )=q}, ∀q∈ρ−1((r0,r1)) .
The mapping is analytic thanks to the analyticity of the vector fieldh. The compact K canE obviously be chosen semi-analytic. The proposition follows now from [25, Prop. 1.3.7].
3. Compactness
Let✁ ⊂ L
k
2[0,1] be the domain of the endpoint mapping f . Recall that✁ is a neighborhood of the origin of Lk2[0,1] and f :✁ →M is a smooth mapping. We are going to use not only defined by the norm “strong” topology in the Hilbert space Lk2[0,1], but also weak topology. We denote by✁
weak the topological space defined by weak topology restricted to✁.
PROPOSITION3. f :✁
weak→M is a continuous mapping.
This proposition easily follows from some classical results on the continuous dependence of solutions to ordinary differential equations on the right-hand side. Nevertheless, I give an independent proof in terms of the chronological calculus (see [1, 5]) since it is very short. We have
f(u) = q0−→exp
Z 1
0
k
X
i=1
ui(t)Xidt
= q0+ k
X
i=1
q0
Z 1
0
ui(t)−→exp Z t
0
k
X
j=1
uj(t)Xjdτ
The integration by parts gives:
Z 1
0
ui(t)−→exp
Z t
0
k
X
j=1
uj(t)Xjdτ
dt=
Z 1
0
ui(t)dt−→exp
Z 1
0
k
X
j=1
uj(t)Xjdt −
k
X
i=1
Z 1
0
uj(t)
Z t
0
ui(τ )dτ−→exp
Z t
0
k
X
j=1
uj(t)Xjdτ
dt◦Xj.
It remains to mention that the mapping u(·)7→R0·u(τ )dτis a compact operator in Lk2[0,1]. A
detailed study of the continuity of−→exp in various topologies see in [18].
THEOREM2. The set of minimal geodesics of a prescribed length r is compact in H1
-topology for any small enough r .
Proof. We have to prove that fr−1(S(r))is a compact subset of Ur. First of all, fr−1(S(r))= f−1(S(r))∩conv Ur, where conv Ur is a ball in Lk2[0,1]. This is just because S(r)cannot
be reached by trajectories of the length smaller than r . Then the continuity ofρ implies that S(r)=ρ−1(r)is a closed set and the continuity of f in weak topology implies that f−1(S(r))is weakly closed. Since conv Ur is weakly compact we obtain that fr−1(S(r))is weakly compact.
What remains is to note that weak topology resricted to the sphere Ur in the Hilbert space is
equivalent to strong topology.
THEOREM3. Suppose that all minimal geodesics of the length r are regular. Then we have that −1(S(r))∩H(r)is compact.
Proof. Denote by uψ(0)the extremal control associated withψ (0)∈ H(r)so that (ψ (0))=
f(uψ(0)). We have uψ(0)=(h1(ψ (·)), . . . ,hk(ψ (·)))(see Proposition 1 and its Corollary). In
particular, uψ(0)continuously depends onψ (0).
Take a sequence ψm(0) ∈ −1(S(r))∩H(r), m = 1,2, . . .; the controls uψm(0) are minimal, the set of minimal controls of the length r is compact, hence there exists a convergent subsequence of this sequence of controls and the limit is again a minimal control. To simplify notations, we suppose without losing generality that the sequence uψm(0), m = 1,2, . . ., is already convergent,∃limm→∞uψm(0)= ¯u.
It follows from Proposition 1 thatψm(1)Duψm(0)f =uψm(0). Suppose that M is endowed
with some Riemannian structure so that the length|ψm(1)|of the cotangent vectorψm(1)has a
sense. There are two possibilities: either|ψm(1)| → ∞(m → ∞) orψm(1), m =1,2, . . .,
contains a convergent subsequence.
In the first case we come to the equationλDu¯f = 0, whereλis a limiting point of the sequence |ψm1(1)|ψm(1),|λ| =1. Henceu is an abnormal minimal control that contradicts the¯
assumption of the theorem.
In the second case letψml(1), l =1,2, . . ., be a convergent subsequence. Thenψml(0),
l =1,2, . . ., is also convergent,∃liml→∞ψml(0)= ¯ψ (0)∈ H(r). Thenu¯ =uψ(¯ 0)and we are done.
COROLLARY2. Let M and the sub-Riemannian structure be real-analytic. Suppose that
Compactness for Sub-Riemannian 7
ρ−1((r0,r ]).
Proof. According to Theorem 3, K0 = −1(S(r0))∩H(r0)is a compact set and{uψ(0) :
ψ (0)∈ K0}is the set of all minimal extremal controls of the length r0. The minimality of an extremal control uψ(0)implies the minimality of the control usψ(0)for s<1, since usψ(0)(τ )=
suψ(0)(τ )and a reparametrized piece of a minimal geodesic is automatically minimal. Hence S(r1)⊂
r1
r0K0
for r1≥r0and the required subanalyticity follows from Proposition 2.
Corollary 2 gives a rather strong sufficient condition for subanalyticity of the distance func-tionρout of q0. In particular, the absence of abnormal minimal geodesics implies subanalyticity ofρin a punctured neighborhood of q0. This condition is not however quite satisfactory because
it doesn’t admit abnormal regular geodesics. Though being non generic, abnormal quasi-regular geodesics appear naturally in problems with symmetries. Moreover, they are common in so called nilpotent approximations of sub-Riemannian structures at (see [5, 12]). The nilpotent approximation (or nilpotenization) of a generic sub-Riemannian structure q0leads to a simplified
quasi-homogeneous approximation of the original distance function. It is very unlikely thatρ
loses subanalyticity under the nilpotent approximation, although the above sufficient condition loses its validity. In the next section we give chekable sufficient conditions for subanalyticity, wich are free of the above mentioned defect.
4. Second Variation
Let u∈Ur be an extremal control, i.e. a critical point of fr. Recall that the Hessian of fr at u
is a quadratic mapping
Hesufr : ker Dufr →coker Dufr,
an independent on the choice of local coordinates part of the second derivative of fr at u. Let
(λ, ν)be Lagrange multipliers associated with u so that equation (3) is satisfied. Then the cov-ectorλ: Tf(u)M→ annihilates im Dufrand the composition
λHesufr : ker Dufr →
(5)
is well-defined.
Quadratic form (5) is the second variation of the sub-Riemannian problem at(u, λ, ν). We have
λHesufr(v)=λD2uf(v, v)−ν|v|2, v∈ker Dufr.
Let q(·)be the geodesic associated with the control u. We set
ind(q(·), λ, ν)=ind+(λHesufr)−dim coker Dufr,
(6)
where ind+(λHesufr)is the positive inertia index of the quadratic formλHesufr. Decoding
some of the symbols we can re-write:
ind(q(·), λ, ν) = sup{dim V : V⊂ker Dufr, λDu2f(v, v) > ν|v|2,∀v∈V\0} −dim{λ′∈T∗f(u)M :λ′Dufr =0}.
REMARK2. Index (5) doesn’t depend on the choice of the orthonormal frame X1, . . . ,Xk
and is actually a characteristic of the geodesic q(·)and the Lagrange multipliers(λ, ν). Indeed, a change of the frame leads to a smooth transformation of the Hilbert manifold Urand to a linear
transformation of variables in the quadratic formλHesufr and linear mapping Dufr. Both terms
in the right-hand side of (5) remain unchanged.
PROPOSITION4. (u, λ, ν) 7→ ind(q(·), λ, ν) is a lower semicontinuous function on the space of solutions of (3).
Proof. We have dim coker Dufr = codim ker Dufr. Here ker Dufr = ker Duf ∩ {u}⊥ ⊂ Lk2[0,1] is a subspace of finite codimension in Lk2[0,1]. The multivalued mapping u 7−→
(ker Dufr)∩Ur is upper semicontinuous in the Hausdorff topology, just because u 7→ Duf
is continuous.
Take(u, λ, ν)satisfying (3). If u′is close enough to u, then ker Du′fr is arbitraryly close
to a subspace of codimension
dim coker Dufr −dim coker Du′fr
in Dufr. Suppose V ⊂ ker Dufr is a finite-dimensional subspace such that λHesufrV is
a positive definite quadratic form. If u′ is sufficiently close to u, then ker Du′fr contains a
subspace V′of dimension
dim V−(dim coker Dufr−dim coker Du′fr)
that is arbitrarily close to a subspace of V . Ifλ′is sufficiently close toλ, then the quadratic form
λ′Hesu′fr
V′is positive definite.
We come to the inequality ind(q′(·), λ′, ν′)≥ind(q(·), λ, ν)for any solution(u′, λ′, ν′)of (3) close enough to(u, λ, ν); here q′(·)is the geodesic associated to the control u′.
THEOREM4. If q(·)is minimal geodesic, then there exist associated with q(·)Lagrange multipliersλ, νsuch that ind(q(·), λ, ν) <0.
This theorem is a direct corollary of a general result announced in [2] and proved in [3]; see also [8] for the updated proof of exactly this corollary.
THEOREM5. Suppose that ind(q(·), λ,0)≥0 for any abnormal geodesic q(·)of the length r and associated Lagrange multipliers(λ,0). Then there exists a compact Kr ⊂H(r)such that S(r)= (Kr).
Proof. We use notations introduced in the first paragraph of the proof of Theorem 3. Let qψ(0) be the geodesic associated to the control uψ(0). We set
Kr = {ψ (0)∈H(r)∩ −1(S(r)): ind(qψ(0), ψ (1),1) <0}. (7)
It follows from Theorem 4 and the assumption of Theorem 5 that (Kr)=S(r). What remains
is to prove that Kris compact.
Compactness for Sub-Riemannian 9
suppose without losing generality that the sequence uψm(0), m=1,2, . . ., is already convergent,
∃limm→∞uψm(0)= ¯u.
It follows from Proposition 1 thatψm(1)Duψm(0)f =uψm(0). There are two possibilities:
either|ψm(1)| → ∞(m→ ∞) orψm(1), m=1,2, . . ., contains a convergent subsequence.
In the first case we come to the equationλ¯Du¯f = 0, whereλ¯ is a limiting point of the sequence|ψm1(1)|ψm(1),|¯λ| =1. Lower semicontinuity of ind(q(·), λ, ν)implies the inequality
ind(q¯(·),λ,¯ 0) < 0, whereq¯(·)is the geodesic associated with the controlu. We come to a¯ contradiction with the assumption of the theorem.
In the second case letψml(1), l =1,2, . . ., be a convergent subsequence. Thenψml(0),
l = 1,2, . . ., is also convergent,∃liml→∞ψml(0) = ¯ψ (0) ∈ H(r). Thenu¯ = uψ(¯ 0) and ind(q¯(·),ψ(¯ 1),1) < 0 because of lower semicontinuity of ind(q(·), λ, ν). Henceψ(¯ 0) ∈ Kr
and we are done.
COROLLARY3. Let M and the sub-Riemannian structure be real-analytic. Suppose r0<r
is such that ind(q(·), λ,0)≥0 for any abnormal geodesic q(·)of the length r0and associated
Lagrange multipliers(λ,0). Thenρis subanalytic onρ−1((r0,r ]).
Proof. Let Kr0 be defined as in (7). Then Kr0 is compact and{uψ(0) : ψ (0) ∈ Kr0}is the
set of all minimal extremal controls of the length r0. The minimality of an extremal control
uψ(0) implies the minimality of the control usψ(0)for s < 1, since usψ(0)(τ ) = suψ(0)(τ ) and a reparametrized piece of a minimal geodesic is automatically minimal. Hence S (r1) ⊂
r1
r0Kr0
for r1≥r0and the required subanalyticity follows from Proposition 2.
Among 2 terms in expression (6) for ind(q(·), λ, ν)only the first one, the inertia index of the second variation, is nontrivial to evaluate. Fortunately, there is an efficient way to compute this index for both regular and singular (abnormal) geodesics, as well as a good supply of conditions that garantee the finiteness or infinity of the index (see [2, 4, 6, 9]). The simplest one is the Goh condition (see [6]):
If ind(q(·), ψ (1),0) <+∞, thenψ (t)annihilates12q(t), ∀t∈[0,1].
Recall thatψ (t)annihilates1q(t), 0≤t ≤1, for any Lagrange multiplier(ψ (1),0)associated with q(·). We say that q(·)is a Goh geodesic if there exist Lagrange multipliers(ψ (1),0)such thatψ (t)annihilates12q(t),∀t∈[0,1]. In particuar, strictly abnormal minimal geodesics must be Goh geodesics. Besides that, the Goh condition and Corollary 3 imply
COROLLARY4. Let M and the sub-Riemannian structure be real-analytic and r0<r . If
there are no Goh geodesics of the length r0, thenρis subanalytic onρ−1((r0,r ]).
I’ll finish the paper with a brief analysis of the Goh condition. Suppose that q(·)is an abnormal geodesic with Lagrange multipliers(ψ (1),0), and k = 2. Differentiating the iden-tities h1( ψ (t) ) = h2( ψ (t) ) = 0 with respect to t , we obtain u2(t){h2,h1}( ψ (t) ) =
u1(t){h1,h2}(ψ (t))=0, where{h1,h2}(ψ (t))= hψ (t),[X1,X2](q(t))iis the Poisson bracket.
notation. Takeλ∈T∗M and set
b0(λ)= {h1,h2}(λ),{h1,h3}(λ), . . . ,{hk−1,hk}(λ),
a vector in k(k2−1) whose coordinates are numbers{hi,hj}(λ), 1 ≤ i < j ≤k, with
lexico-graphically ordered indeces(i,j). Set alsoβ0 = k(k2−1). The Goh condition for q(·), ψ (1)
implies the identity b0(ψ (t))=0,∀t∈[0,1]. The differentiation of this identity with respect to
t in virtue of (4) gives the equality
k
X
i=1
ui(t){hi,b0}(ψ (t))=0, 0≤t≤1.
(8)
Consider the spaceVk β0, the k-th exterior power of β0. The standard lexicographic basis in
Vk β0 gives the identificationVk β0 ∼=
β0
k
. We setβ1=β+
β0
k
and
b1(λ)=(b0(λ),{h1,b0}(λ)∧ · · · ∧ {hk,b0}(λ))∈ β1.
Equality (8) implies: b1(ψ (t))=0, 0≤t≤1.
Now we set by inductionβi+1 = βi +
βi k
, i = 0,1,2, . . ., and fix identifications βi ×
βi k
∼
= βi+1. Finaly, we define
bi+1(λ)=(bi(λ),{h1,bi}(λ)∧ · · · ∧ {hk,bi}(λ))∈ βi+1, i=1,2, . . . .
Successive differentiations of the Goh condition give the equations bi(ψ (t))=0, i =1,2, . . ..
It is easy to check that the equation bi+1(λ)=0 is not, in general, a consequence of the equation bi(λ) = 0 and we indeed impose more and more restrictive conditions on the locus of Goh
geodesics.
A natural conjecture is that admitting Goh geodesics distributions of rank k>2 form a set of infinite codimension in the space of all rank k distributions, i.e. they do not appear in generic smooth families of distributions parametrized by finite-dimensional manifolds. It may be not technically easy, however, to turn this conjecture into the theorem.
Anyway, Goh geodesics are very exclusive for the distributions of rank greater than 2. Yet they may become typical under a priori restictions on the growth vector of the distribution (see [6]).
Note in proof. An essential progress was made while the paper was waiting for the publication. In particular, the conjecture on Goh geodesics has been proved as well as the conjecture stated at the end of the Introduction. These and other results will be included in our joined paper with Jean Paul Gauthier, now in preparation.
References
Compactness for Sub-Riemannian 11
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AMS Subject Classification: ???.
Andrei AGRACHEV
Steklov Mathematical Institute, ul. Gubkina 8,
Moscow 117966, Russia & S.I.S.S.A.
Rend. Sem. Mat. Univ. Pol. Torino Vol. 56, 4 (1998)
M. Bardi
∗– S. Bottacin
ON THE DIRICHLET PROBLEM
FOR NONLINEAR DEGENERATE ELLIPTIC EQUATIONS
AND APPLICATIONS TO OPTIMAL CONTROL
Abstract.
We construct a generalized viscosity solution of the Dirichlet problem for fully nonlinear degenerate elliptic equations in general domains by the Perron-Wiener-Brelot method. The result is designed for the Hamilton-Jacobi-Bellman-Isaacs equations of time-optimal stochastic control and differential games with discon-tinuous value function. We study several properties of the generalized solution, in particular its approximation via vanishing viscosity and regularization of the do-main. The connection with optimal control is proved for a deterministic minimum-time problem and for the problem of maximizing the expected escape minimum-time of a degenerate diffusion process from an open set.
Introduction
The theory of viscosity solutions provides a general framework for studying the partial differ-ential equations arising in the Dynamic Programming approach to deterministic and stochastic optimal control problems and differential games. This theory is designed for scalar fully nonlin-ear PDEs
F(x,u(x),Du(x),D2u(x))=0 in ,
(1)
whereis a general open subset of N, with the monotonicity property F(x,r,p,X)≤F(x,s,p,Y)
if r≤s and X−Y is positive semidefinite,
(2)
so it includes 1st order Hamilton-Jacobi equations and 2nd order PDEs that are degenerate elliptic or parabolic in a very general sense [18, 5].
The Hamilton-Jacobi-Bellman (briefly, HJB) equations in the theory of optimal control of diffusion processes are of the form
sup α∈A
αu
=0,
(3)
∗* Partially supported by M.U.R.S.T., projects “Problemi nonlineari nell’analisi e nelle applicazioni
fisiche, chimiche e biologiche” and “Analisi e controllo di equazioni di evoluzione deterministiche e stocas-tiche”, and by the European Community, TMR Network “Viscosity solutions and their applications”.
whereαis the control variable and, for eachα, αis a linear nondivergence form operator αu :
= −aαi j ∂
2u
∂xi∂xj + bαi ∂u
∂xi +
cαu− fα,
(4)
where f and c are the running cost and the discount rate in the cost functional, b is the drift of the system, a= 1
2σ σ
T andσ is the variance of the noise affecting the system (see Section 3.2).
These equations satisfy (2) if and only if
aαi j(x)ξiξj≥0 and cα(x)≥0, for all x∈, α∈ A, ξ∈ N,
(5)
and these conditions are automatically satisfied by operators coming from control theory. In the case of deterministic systems we have ai jα ≡ 0 and the PDE is of 1st order. In the theory of two-person zero-sum deterministic and stochastic differential games the Isaacs’ equation has the form
sup α∈A
inf β∈B
α,β
u=0,
(6)
whereβis the control of the second player and α,β are linear operators of the form (4) and satisfying assumptions such as (5).
For many different problems it was proved that the value function is the unique continuous viscosity solution satisfying appropriate boundary conditions, see the books [22, 8, 4, 5] and the references therein. This has a number of useful consequences, because we have PDE methods available to tackle several problems, such as the numerical calculation of the value function, the synthesis of approximate optimal feedback controls, asymptotic problems (vanishing noise, penalization, risk-sensitive control, ergodic problems, singular perturbations. . .). However, the theory is considerably less general for problems with discontinuous value function, because it is restricted to deterministic systems with a single controller, where the HJB equation is of first order with convex Hamiltonian in the p variables. The pioneering papers on this issue are due to Barles and Perthame [10] and Barron and Jensen [11], who use different definitions of non-continuous viscosity solutions, see also [27, 28, 7, 39, 14], the surveys and comparisons of the different approaches in the books [8, 4, 5], and the references therein.
For cost functionals involving the exit time of the state from the set, the value function is discontinuous if the noise vanishes near some part of the boundary and there is not enough controllability of the drift; other possible sources of discontinuities are the lack of smoothness of∂, even for nondegenerate noise, and the discontinuity or incompatibility of the boundary data, even if the drift is controllable (see [8, 4, 5] for examples). For these functionals the value should be the solution of the Dirichlet problem
F(x,u,Du,D2u)=0 in ,
u=g on∂ ,
(7)
where g(x)is the cost of exitingat x and we assume g∈C(∂). For 2nd order equations, or 1st order equations with nonconvex Hamiltonian, there are no local definitions of weak solution and weak boundary conditions that ensure existence and uniqueness of a possibly discontinuous solution. However a global definition of generalized solution of (7) can be given by the following variant of the classical Perron-Wiener-Brelot method in potential theory. We define
:= {w∈ BU SC()subsolution of (1), w≤g on∂}
✁
On the Dirichelet problem 15
where BU SC() (respectively, B L SC()) denote the sets of bounded upper (respectively, lower) semicontinuous functions on, and we say that u :→ is a generalized solution of (7) if
u(x)= sup w∈
w(x)= inf
W∈✁
W(x) .
(8)
With respect to the classical Wiener’s definition of generalized solution of the Dirichlet problem for the Lapalce equation in general nonsmooth domains [45] (see also [16, 26]), we only replace sub- and superharmonic functions with viscosity sub- and supersolutions. In the classical theory the inequality supw∈ w≤infW∈✁W comes from the maximum principle, here it comes from the Comparison Principle for viscosity sub- and supersolutions; this important result holds under some additional assumptions that are very reasonable for the HJB equations of control theory, see Section 1.1; for this topic we refer to Jensen [29] and Crandall, Ishii and Lions [18]. The main difference with the classical theory is that the PWB solution for the Laplace equation is harmonic inand can be discontinuous only at boundary points where∂is very irregular, whereas here u can be discontinuous also in the interior and even if the boundary is smooth: this is because the very degenerate ellipticity (2) neither implies regularizing effects, nor it guarantees that the boundary data are attained continuously. Note that if a continuous viscosity solution of (7) exists it coincides with u, and both the sup and the inf in (8) are attained.
Perron’s method was extended to viscosity solutions by Ishii [27] (see Theorem 1), who used it to prove general existence results of continuous solutions. The PWB generalized solution of (7) of the form (8) was studied indipendently by the authors and Capuzzo-Dolcetta [4, 1] and by M. Ramaswamy and S. Ramaswamy [38] for some special cases of equations of the form (1), (2). In [4] this notion is called envelope solution and several properties are studied, in particular the equivalence with the generalized minimax solution of Subbotin [41, 42] and the connection with deterministic optimal control. The connection with pursuit-evasion games can be found in [41, 42] within the Krasovskii-Subbotin theory, and in our paper with Falcone [3] for the Fleming value; in [3] we also study the convergence of a numerical scheme.
The purposes of this paper are to extend the existence and basic properties of the PWB solution in [4, 1, 38] to more general operators, to prove some new continuity properties with respect to the data, in particular for the vanishing viscosity method and for approximations of the domain, and finally to show a connection with stochastic optimal control. For the sake of completeness we give all the proofs even if some of them follow the same argument as in the quoted references.
Let us now describe the contents of the paper in some detail. In Subsection 1.1 we recall some known definitions and results. In Subsection 1.2 we prove the existence theorem under an assumption on the boundary data g that is reminiscent of the compatibility conditions in the theory of 1st order Hamilton-Jacobi equations [34, 4]; this condition implies that the PWB solution is either the minimal supersolution or the maximal subsolution (i.e., either the inf or the sup in (8) is attained), and it is verified in time-optimal control problems. We recall that the classical Wiener Theorem asserts that for the Laplace equation any continuous boundary function g is resolutive (i.e., the PWB solution of the corresponding Dirichlet problem exists), and this was extended to some quasilinear nonuniformly elliptic equations, see the book of Heinonen, Kilpel¨ainen and Martio [25]. We do not know at the moment if this result can be extended to some class of fully nonlinear degenerate equations; however we prove in Subsection 2.1 that the set of resolutive boundary functions in our context is closed under uniform convergence as in the classical case (cfr. [26, 38]).
in the weak viscosity sense [10, 28, 18, 8, 4]. Subsection 2.1 is devoted to the stability of the PWB solution with respect to the uniform convergence of the boundary data and the operator F. In Subsection 2.2 we consider merely local uniform perturbations of F, such as the vanishing viscosity, and prove a kind of stability provided the setis simultaneously approximated from the interior.
In Subsection 2.3 we prove that for a nested sequence of open subsetsnofsuch that
S
nn = , if unis the PWB solution of the Dirichlet problem inn, the solution u of (7)
satisfies
u(x)=lim
n un(x) , x∈ .
(9)
This allows to approximate u with more regular solutions unwhen∂is not smooth andnare
chosen with smooth boundary. This approximation procedure goes back to Wiener [44] again, and it is standard in elliptic theory for nonsmooth domains where (9) is often used to define a generalized solution of (7), see e.g. [30, 23, 12, 33]. In Subsection 2.3 we characterize the boundary points where the data are attained continuously in terms of the existence of suitable local barriers.
The last section is devoted to two applications of the previous theory to optimal control. The first (Subsection 3.1) is the classical minimum time problem for deterministic nonlinear systems with a closed target. In this case the lower semicontinuous envelope of the value function is the PWB solution of the homogeneous Dirichlet problem for the Bellman equation. The proof we give here is different from the one in [7, 4] and simpler. The second application (Subsection 3.2) is about the problem of maximizing the expected discounted time that a controlled degenerate diffusion process spends in. Here we prove that the value function itself is the PWB solution of the appropriate problem. In both cases g≡0 is a subsolution of the Dirichlet problem, which implies that the PWB solution is also the minimal supersolution.
It is worth to mention some recent papers using related methods. The thesis of Bettini [13] studies upper and lower semicontinuous solutions of the Cauchy problem for degenerate parabolic and 1st order equations with applications to finite horizon differential games. Our paper [2] extends some results of the present one to boundary value problems where the data are prescribed only on a suitable part of∂. The first author, Goatin and Ishii [6] study the boundary value problem for (1) with Dirichlet conditions in the viscosity sense; they construct a PWB-type generalized solution that is also the limit of approximations offrom the outside, instead of the inside. This solution is in general different from ours and it is related to control problems involving the exit time from, instead of.
1. Generalized solutions of the Dirichlet problem
1.1. Preliminaries
Let F be a continuous function
F :× × N ×S(N)→ ,
whereis an open subset of N, S(N)is the set of symmetric N×N matrices equipped with its usual order, and assume that F satisfies (2). Consider the partial differential equation
F(x,u(x),Du(x),D2u(x))=0 in ,
On the Dirichelet problem 17
where u :→ , Du denotes the gradient of u and D2u denotes the Hessian matrix of second derivatives of u. From now on subsolutions, supersolutions and solutions of this equation will be understood in the viscosity sense; we refer to [18, 5] for the definitions. For a general subset E of Nwe indicate with U SC(E), respectively L SC(E), the set of all functions E → upper, respectively lower, semicontinuous, and with BU SC(E), B L SC(E)the subsets of functions that are also bounded.
DEFINITION1. We will say that equation (10) satisfies the Comparison Principle if for all subsolutionsw∈ BU SC()and supersolutions W ∈ B L SC()of (10) such thatw≤W on
∂, the inequalityw≤W holds in.
We refer to [29, 18] for the strategy of proof of some comparison principles, examples and references. Many results of this type for first order equations can be found in [8, 4].
The main examples we are interested in are the Isaacs equations:
sup α
inf β
α,βu(x)
=0 (11)
and
inf β supα
α,βu(x)=0, (12)
where
α,βu(x)= −aα,β
i j (x)
∂2u
∂xi∂xj +
bα,βi (x)∂u ∂xi +
cα,β(x)u− fα,β(x) .
Here F is
F(x,r,p,X)=sup α
inf
β{−trace(a
α,β(x)X)
+bα,β(x)·p+cα,β(x)r− fα,β(x)}.
If, for all x∈, aα,β(x)= 1 2σ
α,β(x)(σα,β(x))T, whereσα,β(x)is a matrix of order N×M,T
denotes the transpose matrix,σα,β,bα,β,cα,β, fα,βare bounded and uniformly continuous in
, uniformly with respect toα, β, then F is continuous, and it is proper if in addition cα,β≥0 for allα, β.
Isaacs equations satisfy the Comparison Principle ifis bounded and there are positive constants K1,K2, and C such that
F(x,t,p,X)−F(x,s,q,Y)≤max{K1trace(Y−X), K1(t−s)} +K2|p−q|,
(13)
for all Y ≤X and t≤s,
kσα,β(x)−σα,β(y)k ≤ C|x−y|, for all x,y∈and allα, β
(14)
|bα,β(x)−bα,β(y)| ≤ C|x−y|, for all x,y∈and allα, β ,
(15)
see Corollary 5.11 in [29]. In particular condition (13) is satisfied if and only if
max{λα,β(x),cα,β(x)} ≥K >0 for all x∈, α∈A, β∈ B,
Given a function u :→[−∞,+∞], we indicate with u∗and u∗, respectively, the upper
and the lower semicontinuous envelope of u, that is,
u∗(x) := lim
rց0sup{u(y): y∈,|y−x| ≤r},
u∗(x) := lim
rց0inf{u(y): y∈, |y−x| ≤r}.
PROPOSITION1. Let S (respectively Z ) be a set of functions such that for allw∈ S (re-spectively W ∈ Z )w∗is a subsolution (respectively W∗is a supersolution) of (10). Define the
function
u(x):= sup w∈S
w(x), x∈, (respectively u(x):= inf
W∈ZW(x)) .
If u is locally bounded, then u∗is a subsolution (respectively u∗is a supersolution) of (10).
The proof of Proposition 1 is an easy variant of Lemma 4.2 in [18].
PROPOSITION2. Letwn ∈ BU SC()be a sequence of subsolutions (respectively Wn ∈ B L SC()a sequence of supersolutions) of (10), such thatwn(x)ցu(x)for all x∈ (respec-tively Wn(x)րu(x)) and u is a locally bounded function. Then u is a subsolution (respectively supersolution) of (10).
For the proof see, for instance, [4]. We recall that, for a generale subset E of Nandxˆ ∈E , the second order superdifferential of u atx is the subset Jˆ E2,+u(xˆ)of N ×S(N)given by the pairs(p,X)such that
u(x)≤u(xˆ)+p·(x− ˆx)+1
2X(x− ˆx)·(x− ˆx)+o(|x− ˆx|
2)
for E ∋ x → ˆx . The opposite inequality defines the second order subdifferential of u atx ,ˆ JE2,−u(xˆ).
LEMMA1. Let u∗be a subsolution of (10). If u∗fails to be a supersolution at some point ˆ
x∈, i.e. there exist(p,X)∈J2,−u∗(xˆ)such that
F(xˆ,u∗(xˆ),p,X) <0,
then for all k> 0 small enough, there exists Uk :→ such that Uk∗is subsolution of (10) and
Uk(x)≥u(x), sup(Uk−u) >0,
Uk(x)=u(x)for all x∈such that|x− ˆx| ≥k.
The proof is an easy variant of Lemma 4.4 in [18]. The last result of this subsection is Ishii’s extension of Perron’s method to viscosity solutions [27].
THEOREM1. Assume there exists a subsolution u1and a supersolution u2of (10) such that
u1≤u2, and consider the functions
U(x) := sup{w(x): u1≤w≤u2, w∗subsolution of(10)},
W(x) := inf{w(x): u1≤w≤u2, w∗supersolution of(10)}.
On the Dirichelet problem 19
1.2. Existence of solutions by the PWB method
In this section we present a notion of weak solution for the boundary value problem
F(x,u,Du,D2u)=0 in ,
u=g on∂ ,
(16)
where F satisfies the assumptions of Subsection 1.1 and g :∂→ is continuous. We recall that ,
✁
are the sets of all subsolutions and all supersolutions of (16) defined in the Introduction.
DEFINITION2. The function defined by
Hg(x):= sup w∈
w(x) ,
is the lower envelope viscosity solution, or Perron-Wiener-Brelot lower solution, of (16). We will refer to it as the lower e-solution. The function defined by
Hg(x):= inf W∈✁
W(x) ,
is the upper envelope viscosity solution, or PWB upper solution, of (16), briefly upper e-solution. If Hg= Hg, then
Hg:=Hg=Hg
is the envelope viscosity solution or PWB solution of (16), briefly e-solution. In this case the data g are called resolutive.
Observe that Hg≤Hgby the Comparison Principle, so the e-solution exists if the
inequal-ity≥holds as well. Next we prove the existence theorem for e-solutions, which is the main result of this section. We will need the following notion of global barrier, that is much weaker than the classical one.
DEFINITION3. We say thatwis a lower (respectively, upper) barrier at a point x∈∂if
w∈ (respectively,w∈✁
) and
lim
y→xw(y)=g(x) .
THEOREM2. Assume that the Comparison Principle holds, and that ,✁
are nonempty. i) If there exists a lower barrier at all points x ∈∂, then Hg=minW∈✁W is the e-solution
of (16).
ii) If there exists an upper barrier at all points x∈∂, then Hg=maxw∈ wis the e-solution
of (16).
Proof. Letwbe the lower barrier at x ∈∂, then by definitionw≤Hg. Thus
(Hg)∗(x)=lim infy→x Hg(y)≥lim infy→x w(y)=g(x) .
By Theorem 1(Hg)∗is a supersolution of (10), so we can conclude that(Hg)∗ ∈
✁ . Then
(Hg)∗≥Hg≥Hg, so Hg=Hgand Hg∈
EXAMPLE1. Consider the problem
−ai j(x)uxixj(x)+bi(x)uxi(x)+c(x)u(x)=0 in ,
u(x)=g(x) on∂ ,
(17)
with the matrix ai j(x)such that a11(x) ≥ µ > 0 for all x ∈ . In this case we can show
that all continuous functions on∂are resolutive. The proof follows the classical one for the Laplace equation, the only hard point is checking the superposition principle for viscosity sub-and supersolutions. This can be done by the same methods sub-and under the same assumptions as the Comparison Principle.
1.3. Consistency properties and examples
Next results give a characterization of the e-solution as pointwise limit of sequences of sub and supersolutions of (16). If the equation (10) is of first order, this property is essentially Subbotin’s definition of (generalized) minimax solution of (16) [41, 42].
THEOREM3. Assume that the Comparison Principle holds, and that ,✁
are nonempty. i) If there exists u∈ continuous at each point of∂and such that u=g on∂, then there
exists a sequencewn∈ such thatwnրHg. ii) If there exists u∈✁
continuous at each point of∂and such that u=g on∂, then there exists a sequence Wn∈
✁
such that WnցHg.
Proof. We give the proof only for i), the same proof works for ii). By Theorem 2 Hg =
minW∈✁W . Givenǫ >0 the function
uǫ(x):=sup{w(x):w∈ , w(x)=u(x)if dist(x, ∂) < ǫ}, (18)
is bounded, and uδ≤uǫforǫ < δ. We define
V(x):= lim
n→∞(u1/n)∗(x) ,
and note that, by definition, Hg ≥ uǫ ≥ (uǫ)∗, and then Hg ≥ V . We claim that(uǫ)∗is
supersolution of (10) in the set
ǫ:= {x∈: dist(x, ∂) > ǫ}.
To prove this claim we assume by contradiction that(uǫ)∗fails to be a supersolution at y∈ǫ. Note that, by Proposition 1,(uǫ)∗is a subsolution of (10). Then by Lemma 1, for all k > 0 small enough, there exists Uksuch that Uk∗is subsolution of (10) and
sup
(Uk−uǫ) >0, Uk(x)=uǫ(x)if|x−y| ≥k. (19)
We fix k≤dist(y, ∂)−ǫ, so that Uk(x)=uǫ(x)=u(x)for all x such that dist(x, ∂) < ǫ. Then Uk∗(x)=u(x), so Uk∗∈ and by the definition of uǫ we obtain Uk∗≤uǫ. This gives a contradiction with (19) and proves the claim.
By Proposition 2 V is a supersolution of (10) in. Moreover if x ∈ ∂, for allǫ >0,
(uǫ)∗(x)= g(x), because uǫ(x)=u(x)if dist(x, ∂) < ǫby definition, u is continuous and
On the Dirichelet problem 21
To complete the proof we definewn:= (u1/n)∗, and observe that this is a nondecreasing
sequence in whose pointwise limit is≥V by definition of V . On the other handwn≤Hgby
definition of Hg, and we have shown that Hg=V , sownրHg.
COROLLARY1. Assume the hypotheses of Theorem 3. Then Hgis the e-solution of (16 if and only if there exist two sequences of functionswn∈ , Wn∈
✁
, such thatwn=Wn=g on
∂and for all x∈
wn(x)→Hg(x), Wn(x)→Hg(x)as n→ ∞.
REMARK1. It is easy to see from the proof of Theorem 3, that in case i), the e-solution Hg
satisfies
Hg(x)=sup
ǫ uǫ(x) x∈ , where
uǫ(x):=sup{w(x):w∈ , w(x)=u(x)for x∈\2ǫ}, (20)
and2ǫ, ǫ ∈]0,1], is any family of open sets such that2ǫ ⊆ ,2ǫ ⊇ 2δ forǫ < δand S
ǫ2ǫ=.
EXAMPLE2. Consider the Isaacs equation (11) and assume the sufficient conditions for the Comparison Principle.
• If
g≡0 and fα,β(x)≥0 for all x∈, α∈A, β∈B,
then u≡0 is subsolution of the PDE, so the assumption i)of Theorem 3 is satisfied. • If the domainis bounded with smooth boundary and there existα∈A andµ >0 such
that
aα,βi j (x)ξiξj≥µ|ξ|2for allβ∈ B, x∈, ξ∈ N,
then there exists a classical solution u of (
inf β∈B
α,βu=0 in ,
u=g on∂ ,
see e.g. Chapt. 17 of [24]. Then u is a supersolution of (11), so the hypothesis ii)of Theorem 3 is satisfied.
Next we compare e-solutions with Ishii’s definitions of non-continuous viscosity solution and of boundary conditions in viscosity sense. We recall that a function u∈BU SC() (respec-tively u ∈ B L SC()) is a viscosity subsolution (respectively a viscosity supersolution) of the boundary condition
u=g or F(x,u,Du,D2u)=0 on∂ ,
if for all x∈∂andφ∈C2()such that u−φattains a local maximum (respectively minimum) at x, we have
(u−g)(x)≤0(resp. ≥0)or F(x,u(x),Dφ (x),D2φ (x))≤0(resp. ≥0) .
An equivalent definition can be given by means of the semijets J2,+ u(x), J
2,−
u(x)instead of the test functions, see [18].
PROPOSITION3. If Hg : → is the lower e-solution (respectively, Hg is the upper e-solution) of (16), then H∗gis a subsolution (respectively, Hg∗is a supersolution) of (10) and of the boundary condition (21).
Proof. If Hg is the lower e-solution, then by Proposition 1, H∗g is a subsolution of (10). It remains to check the boundary condition.
Fix an y∈∂such that H∗g(y) >g(y), andφ∈C2()such that H∗g−φattains a local maximum at y. We can assume, without loss of generality, that
H∗g(y)=φ (y), (H∗g−φ)(x)≤ −|x−y|3for all x∈∩B(y,r) .
By definition of H∗g, there exists a sequence of points xn→y such that
(Hg−φ)(xn)≥ −
1
nfor all n.
Moreover, since Hgis the lower e-solution, there exists a sequence of functionswn ∈ S such
that
Hg(xn)−
1
n < wn(xn)for all n.
Since the functionwn−φis up
